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The recommendation of new plant varieties for commercial use requires reliable and accurate predictions of the average yield of each variety across a range of target environments and knowledge of important interactions with the environment. This information is obtained from series of plant variety trials, also known as multi-environment trials (MET). Cullis, Gogel, Verbyla, and Thompson (1998) presented a spatial mixed model approach for the analysis of MET data. In this paper we extend the analysis to include multiplicative models for the variety effects in each environment. The multiplicative model corresponds to that used in the multivariate technique of factor analysis. It allows a separate genetic variance for each environment and provides a parsimonious and interpretable model for the genetic covariances between environments. The model can be regarded as a random effects analogue of AMMI (additive main effects and multiplicative interactions). We illustrate the method using a large set of MET data from a South Australian barley breeding program.

After estimation of e ects from a linear mixed model, it is often useful to form predicted values for certain factor/variate combinations. This process has been well-deÿned for linear models, but the introduction of random e ects means that a decision has to be made about the inclusion or exclusion of random model terms from the predictions, including the residual error. For spatially correlated data, kriging then becomes prediction from the ÿtted model. In many cases, the size of the matrices required to calculate predictions and their covariance matrix directly can be prohibitive. An e cient computational strategy for calculating predictions and their standard errors is given, which includes the ability to detect the invariance of predictions to the parameterisation used in the model.

The analysis of series of crop variety trials has a long history with the earliest approaches being based on ANOVA methods. Kempton (1984) discussed the inadequacies of this approach, summarized the alternatives available at that time and noted that all of these approaches could be classified as multiplicative models. Recently, mixed model approaches have become popular for the analysis of series of variety trials. There are numerous reasons for their use, including the ease with which incomplete data (not all varieties in all trials) can be handled and the ability to appropriately model within-trial error variation. Currently, the most common mixed model approaches for series of variety trials are mixed model versions of the methods summarized by Kempton (1984). In the present paper a general formulation that encompasses all of these methods is described, then individual methods are considered in detail.

An algorithm is described to estimate variance components for a univariate animal model using REML. Sparse matrix techniques are employed to calculate those elements of the inverse of the coefficient matrix required for the first derivatives of the likelihood. Residuals and fitted values for random effects can be used to derive additional right-hand sides for which the mixed model equations can be repeatedly solved in turn to yield an average of the observed and expected second derivatives of the likelihood function.This Newton method, using average information, generally converges in 4 0 iterations. Although the time required per iteration is two to three times greater than that required per likelihood evaluation for derivative-free methods, the total time to convergence is generally much less. An example of a complex model, involving correlated direct and maternal genetic effects, and an additional uncorrelated random effect, indicates that REML, using average information, is about five times faster than a derivativefree algorithm, using the simplex method, which is about three times faster than an expectation-maximization algorithm.

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